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Additional Mathematics
Project Work 2011
Name : Ng Ken Jie
Class : 5S5
I.C. Number : 941020-14-6571
School : SMJK Chong Hwa
Angka Giliran : 22
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Content Page
No. Content Page
1. Introduction 1 -2
2. Part I 3 -4
3. Part II 5 - 13
4. Part III 14 - 16
5. Further Exploration 17 18
6. Reflection 19 -20
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Introduction
The circle has been known since before the beginning of recorded history. It is the
basis for the wheel, which, with related inventions such as gears, makes much
of modern civilization possible. In mathematics, the study of the circle has helped
inspire the development of geometry and calculus. Circles had been used in daily
lives to help people in their living. There are a lot of things around us that are related to
circles or parts of a circle.
A circle is a simple shape ofEuclidean geometry consisting of the set ofpoints in
a plane that are a given distance from a given point, the centre. The distance
between any of the points and the centre is called the radius.
Circles are simple closed curves which divide the plane into two regions:
an interior and an exterior. In everyday use, the term "circle" may be used
interchangeably to refer to either the boundary of the figure, or to the whole figure
including its interior; in strict technical usage, the circle is the former and the latter is
called a disk.
A circle is a special ellipse in which the two foci are coincident and the eccentricity is
0. Circles are conic sections attained when a right circular cone is intersected by a
plane perpendicular to the axis of the cone.
Circle illustration showing a radius, a diameter, the centre and the circumference
1
http://en.wikipedia.org/wiki/Shapehttp://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Point_(geometry)http://en.wikipedia.org/wiki/Plane_(mathematics)http://en.wikipedia.org/wiki/Centre_(geometry)http://en.wikipedia.org/wiki/Radiushttp://en.wikipedia.org/wiki/Curvehttp://en.wikipedia.org/wiki/Plane_(mathematics)http://en.wikipedia.org/wiki/Interior_(topology)http://en.wikipedia.org/wiki/Disk_(mathematics)http://en.wikipedia.org/wiki/Ellipsehttp://en.wikipedia.org/wiki/Focus_(geometry)http://en.wikipedia.org/wiki/Eccentricity_(mathematics)http://en.wikipedia.org/wiki/Conic_sectionhttp://en.wikipedia.org/wiki/Conical_surfacehttp://en.wikipedia.org/wiki/Conical_surfacehttp://en.wikipedia.org/wiki/Conic_sectionhttp://en.wikipedia.org/wiki/Eccentricity_(mathematics)http://en.wikipedia.org/wiki/Focus_(geometry)http://en.wikipedia.org/wiki/Ellipsehttp://en.wikipedia.org/wiki/Disk_(mathematics)http://en.wikipedia.org/wiki/Interior_(topology)http://en.wikipedia.org/wiki/Plane_(mathematics)http://en.wikipedia.org/wiki/Curvehttp://en.wikipedia.org/wiki/Radiushttp://en.wikipedia.org/wiki/Centre_(geometry)http://en.wikipedia.org/wiki/Plane_(mathematics)http://en.wikipedia.org/wiki/Point_(geometry)http://en.wikipedia.org/wiki/Euclidean_geometryhttp://en.wikipedia.org/wiki/Shape8/2/2019 Additional Mathematics Project 2010
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History
The word "circle" derives from the Greek, kirkos "a circle," from the base ker- which
means to turn or bend. The origins of the words "circus" and "circuit" are closely
related.
The circle has been known since before the beginning of recorded history. Natural
circles would have been observed, such as the Moon, Sun, and a short plant stalk
blowing in the wind on sand, which forms a circle shape in the sand. The circle is the
basis for the wheel, which, with related inventions such as gears, makes much of
modern civilization possible. In mathematics, the study of the circle has helped
inspire the development of geometry, astronomy, and calculus.
Early science, particularly geometry and astrology and astronomy, was connected to
the divine for most medieval scholars, and many believed that there was something
intrinsically "divine" or "perfect" that could be found in circles.
Some highlights in the history of the circle are:
1700 BC The Rhind papyrus gives a method to find the area of a circular field.The result corresponds to 256/81 (3.16049...) as an approximate value of.
300 BC Book 3 ofEuclid's Elements deals with the properties of circles. In Plato's Seventh Letter there is a detailed definition and explanation of the
circle. Plato explains the perfect circle, and how it is different from any drawing,
words, definition or explanation.
1880 Lindemann proves that is transcendental, effectively settling themillennia-old problem ofsquaring the circle.
[2]
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http://en.wikipedia.org/wiki/Circushttp://en.wikipedia.org/wiki/Circuithttp://en.wikipedia.org/wiki/Wheelhttp://en.wikipedia.org/wiki/Gearhttp://en.wikipedia.org/wiki/Astronomyhttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Astrology_and_astronomyhttp://en.wikipedia.org/wiki/History_of_science_in_the_Middle_Ageshttp://en.wikipedia.org/wiki/Rhind_papyrushttp://en.wikipedia.org/wiki/Euclid%27s_Elementshttp://en.wikipedia.org/wiki/Platohttp://en.wikipedia.org/wiki/Seventh_Letterhttp://en.wikipedia.org/wiki/Ferdinand_von_Lindemannhttp://en.wikipedia.org/wiki/Squaring_the_circlehttp://en.wikipedia.org/wiki/Circle#cite_note-1http://en.wikipedia.org/wiki/Circle#cite_note-1http://en.wikipedia.org/wiki/Circle#cite_note-1http://en.wikipedia.org/wiki/Circle#cite_note-1http://en.wikipedia.org/wiki/Squaring_the_circlehttp://en.wikipedia.org/wiki/Ferdinand_von_Lindemannhttp://en.wikipedia.org/wiki/Seventh_Letterhttp://en.wikipedia.org/wiki/Platohttp://en.wikipedia.org/wiki/Euclid%27s_Elementshttp://en.wikipedia.org/wiki/Rhind_papyrushttp://en.wikipedia.org/wiki/History_of_science_in_the_Middle_Ageshttp://en.wikipedia.org/wiki/Astrology_and_astronomyhttp://en.wikipedia.org/wiki/Geometryhttp://en.wikipedia.org/wiki/Sciencehttp://en.wikipedia.org/wiki/Astronomyhttp://en.wikipedia.org/wiki/Gearhttp://en.wikipedia.org/wiki/Wheelhttp://en.wikipedia.org/wiki/Circuithttp://en.wikipedia.org/wiki/Circus8/2/2019 Additional Mathematics Project 2010
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Part 1
Cakes come in variety of forms and flavours and are among favourite desserts served
during special occasions such as birthday parties, Hari Raya, weddings and etc. Cakes
are treasured not only because of their wonderful taste but also in the art of cake
baking and cake decorating. Find out how mathematics is used in cake bakingdecorating and write about your findings
r3
h3
h2
h1
By knowing the value of height, h1 and radius, r, the volume of cake A can be
calculated. This will help the baker to estimate amount of baking powder, flour,
sugar and etc to be used to bake the cake A.
By fixing the r1 value, the baker can identify the types of baking tray can be used
which the diameter of baking tray should be more than 2r1 (the diameter of cake
A)
Therefore, volume of cake B and cake C also can be calculated by using
r2 h2 and
1r32h3 respectively.By calculating the curved surface area at cake A, cake B and cake C by using
formulae 2r1h1 , 2r2h2 and 2r3h3 respectively, the amount of needed cakedecorating can be estimated.
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r1
A
B r2
C
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Q
P
The area of shaded region, P
=
r1
2
r2
2
= (r12 r22)
The area of shaded region, Q
= r22r32
= (r22 r32)
These calculations also can be used to estimate amount of cake decorating needed
or can be used.
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A
B
C
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Part 2
Best Bakery shop received an order from your school to bake a 5 kg of round cake as
shown in Diagram 1 for the Teachers Day celebration.
1) If a kilogram of cake has a volume of 3800cm , and the height of the cake is tobe 7.0cm, calculate the diameter of the baking tray to be used to fit the 5 kg
cake ordered by your school. [Use =3.142]
Volume of 1kg = 3800 cm
Volume of 5kg = 3800 cm x 5
= 19000 cm
Volume of cake =
r
2h
19000 = ()r2(7)r2 = 863.87197r = 29.3917
d = 2r= 58.7834 cm
2) The cake will be baked in an oven with inner dimensions of 80.0cm in length,
60.0cm in width and 45.0 cm in height.
a) If the volume of cake remains the same, explore by using differentvalues of heights, h cm, and the corresponding values of diameters of
the baking tray to be used, dcm. Tabulate your answers
If h = r2(1) = 19000
()r2 = 19000r
2= 6047.1038
r = 77.7631
d = 2r = 2 x 77.7631
d = 155.53 cm
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Height,h (cm) Diameter,d(cm) 1/d2
1.0 155.53 0.41 x 10-5
2.0 109.97 0.83 x 10-5
3.0 89.79 1.240 x 10-4
4.0 77.76 1.653 x 10-4
5.0 69.55 2.067 x 10-4
6.0 63.49 2.480 x 10-4
7.0 58.78 2.894 x 10-4
8.0 54.99 3.306 x 10-4
9.0 51.84 3.721 x 10-4
10.0 49.18 4.134 x 10-4
11.0 46.89 4.548 x 10-4
12.0 44.89 4.962 x 10-4
13.0 43.13 5.375 x 10
-4
14.0 41.56 5.789 x 10
-4
15.0 40.15 6.203 x 10-4
16.0 38.88 6.615 x 10-4
17.0 37.72 7.028 x 10-4
18.0 36.65 7.444 x 10-4
19.0 35.68 7.855 x 10-4
20.0 34.77 8.271 x 10-4
21.0 33.93 8.686 x 10-4
22.0 33.15 9.099 x 10-4
23.0 32.42 9.514 x 10-4
24.0 31.74 9.926 x 10
-4
25.0 31.10 10.33 x 10-4
26.0 30.50 10.74 x 10-4
27.0 29.93 11.16 x 10-4
28.0 29.39 11.57 x 10-4
29.0 28.88 11.98 x 10-4
30.0 28.39 12.10 x 10-4
31.0 27.93 12.81 x 10-4
32.0 27.49 13.23 x 10-4
33.0 27.07 13.64x 10-4
34.0 26.67 14.05 x 10-4
35.0 26.28 14.47 x 10-4
36.0 25.92 14.88 x 10-4
37.0 25.56 15.30 x 10-4
38.0 25.22 15.72 x 10-4
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39.0 24.90 16.12 x 10-4
40.0 24.59 16.55 x 10-4
41.0 24.28 16.96 x 10-4
42.0 23.99 16.01 x 10-4
43.0 23.71 17.78 x 10-4
44.0 23.44 18.20 x 10-4
45.0 23.18 18.61 x 10-4
b) Based on the values in your table,
(i) state the range of heights that is NOT suitable for the cakes and
explain your answers.
The height of the cake cannot be higher than 45 cm and diameter
cannot be more than 60 cm as the cake will be too large to fit into
the oven.
(ii) suggest the dimensions that you think most suitable for the cake.
Give reasons for your answer.
The most suitable dimensions for the cake will be h = 8 cm and d =
54.99 cm so that the cake can fit into the oven and for easyhandling.
c) (i) Form an equation to represent the linear relation between h and d.
Hence, plot a suitable graph based on the equation that you have
formed. [You may draw your graph with the aid of computer
software]
(Please refer to the graph as attached next page)
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(ii) (a) If Best Bakery received an order to bake a cake where the
height of the cake is 10.5 cm, use your graph to determine the
diameter of the round cake pan required.
When h = 10.5 cm, from the graph
1 = 4.4 x 10-4
cm-2
d2
d2
= 1
4.4 x 10-4
d =
-4
= 47.67 cm
(b) If Best Bakery used a 42 cm diameter round cake tray, used
your graph to estimate the height of the cake obtained.
When d = 42 cm,
1 = 1
d2
422
= 5.67 x 10-4
cm-2
From the graph, h = 13.5 cm
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3) Best Bakery has been requested to decorate the cake with fresh cream. The
thickness of the cream is normally set to a uniform layer of about 1 cm.
(a) Estimate the amount of fresh cream required to decorate the cake using
the dimensions that you have suggested in 2(b)(ii)
h = 8cm , d = 54.99
Amount of fresh cream = Volume of fresh cream needed
(area x height)
Amount of fresh cream = Volume of cream at the top surface +
Volume of cream at the side surface
Amount of fresh cream
at top surface
= Area of top surface x height of cream
(3.142)()2 x 12 = 2375 cm
3
Volume of cream at the side surface
= Area of side surface x height of cream
= (circumference of cake x height of cake) x height of cream
= 2(3.142)(54.99/2)(8) x 1
= 1382.23 cm3
Therefore, the amount of fresh cream need to decorate the cake with h =
8cm and d = 54.99,
2375 cm3 + 1382.23 cm3 = 3757.23 cm3
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(b) Suggest three other shapes for cake, that will have the same height and
volume as those suggested in 2(b)(ii). Estimate the amount of fresh cream
to be used on each of the cakes.
i ) T r i a n g l e - s h a p e d b a s e
Base area x height = 19000
Base area = 2375
1/7 x length x width = 2375 length x width = 4750
By trial and improvement
By trial and improvement
4750 = 95 x 50 (length = 95, width = 50)
Slant length of triangle =(95+ 25)= 98.23
Therefore, volume of cream
= area of rectangular front side surface (height of cream)
+ 2(area of slant rectangular both side surface)(height of
cream) + volume of top surface
(Height of cream) + volume of top surface
= (50 x 8)(1) + 2 (98.23x8)(1)+2375 = 4346.68 cm3
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ii) Rectangle shaped-base (cuboid)
Base area x height = 19000
Base area = 19000
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Length x width = 2375
By trial and improvement,
2375 x 50 x 47.5 (length = 50, width = 47.5, height = 8)
Therefore, volume of cream
= 2 (area of left/right side surface)(height of cream)
+ 2(area of front/back side surface)
(Height of cream) + volume of top surface
= 2(8x50)(1) + 2(8 x 47.5)(1) + 2375 = 3935 cm3
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3 . P e n t a g o n - s h a p e d b a s e
Base area x height = 19000
Base area = 2375
= area of 5 similar isosceles
Triangles in a pentagon
Therefore :
5(length x width) = 2375
Length x width = 475
By trial and improvement,
475 x 25 x 19 (length = 25, width = 19)
Therefore, volume of cream
= 5 (area of one rectangular side surface)(height of cream)
+ volume of tope surface
= 5(8x19) + 2375 = 3135 cm3
(c) Based on the values that you have found which shape requires the least
amount of fresh cream to be used?
The Pentagon-shaped cake requires the least amount of fresh cream since
it requires only 3135 cm3
cream
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Part III
Find the dimension of a 5kg round cake that requires the minimum amount of fresh
cream to decorate. Use at least two different methods including Calculus.State whether you choose to bake a cake of such dimensions. Give reasons for your
answers.
Method 1 : Differentiation
Use two equations for this method: the formula for volume of cake (as in Q2/a), and
the formula for amount (volume) of cream to be used for the round cake (as in
Q3/a).
19000 = (3.142)r2h(1)
v = (3.142)r2+2(3.142)rh(2)
From (1):h
= 19000 (3)(3.142)r
2
Sub. (3) in (2):
V = (3.132)r2 + 2(3.142)r ( ( ())V = (3.142)r
2+ ( )
V = (3.142)r2
+ 38000r-1
( )
= 2(3.142)r -
(
)--minium value,
Therefore( ) = 0
( )= 2(3.142)r
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( ()= r
3
6047.104 = r3
r = 18.22
Therefore, h=18.22 cm, d=2r 2(18.22) = 36.44 cm
Method 2: Quadratic Functions
Use the two same equations as in Method 1, but only the formula for amount of
cream is the main equation used as the quadratic function.
f(r) = volume of cream, r = radius of round cake :
19000 = (3.142)r2h(1)
f(r) = (3.142)r2+2(3.142)hr(2)
From (2) :f(r) = (3.142)(r2)(r
2+2hr)--factorize (3.142)
= (3.142)[ ( )2-( )2]--Completing square,with a = (3.142), b=2h and c=0
= (3.142)[r+h)2-h2]
= (3.142)[r+h)2-(3.142)h
2
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(a = (3.142) (positive indicates min. value), min. value = f(r) =(3.142)h,
corresponding value of x = r = --h)
Sub.r = -h into (1) :
19000 = (3.142)(-h)2h
h3
= 6047.104
h = 18.22
Sub.h = 18.22 into (1) :
19000 = (3.142)r2(18.22)
r2
= 331.894
r = 18.22
therefore, h = 18.22 cm, d = 2r = 2(18.22) = 36.44 cm
I would not bake a cake with such dimensions because the dimensions are
not suitable as the height of the cake will be too high and are difficult to
handle. The cake with this height will not be attractive.
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FURTHER EXPLORATION
(a) Find the volume of the first, the second, the third and the fourth cakes. Bycomparing all these values, determine whether the volumes of the cakes form a
number pattern? Explain and elaborate on the number patterns
r = 31 cm, h = 6 cm
Volume of first cake = r2h= 3.142(31)
2(6)
= 18116.772 cm3
Radius of second cake = 31(0.9)
= 27.9
Volume of second cake = r2h= (3.142)(27.9)
2(6)
= 14674.585 cm2
Radius of third cake = 27.9(0.9)
= 25.11Volume of third cake = r2h= (3.142)(25.11)
2(6)
= 11886.414 cm3
Radius of fourth cake = 25.11 (0.9)
= 22.599
Volume of fourth cake = r2h= (3.142)(22.599)
2(6)
= 9627.995 cm3
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Volume of second cake = 14674.585
Volume of first cake 18116.772
= 0.81
= (0.9)2
Volume of third cake = 11886.414Volume of second cake 14674.585
= 0.81
= (0.9)2
Hence,
Volume of second cake
Volume of third cake
Therefore, the number of pattern is based on geometric progression
(b) If the total mass of all the cakes should not exceed 15 kg, calculate the maximum
number of cakes that the bakery needs to bake. Verify your answer using other
methods.
r2h(1-0.9
n)
Sn 1-0.9
15 x 3800 (31)(6)(1-0.9)
0.1
5700 3.142(31)(6)(1-0.9n)
5700
3.142(31)(6) 1-0.9n
0.314626 1-0.9n
0.9n 0.685374
n log 0.9 log 0.685374-0.045757n -0.164072
n 3.586
n 3
The maximum number of cake that the bakery needs to bake is 3
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REFLECTION
While I conducting the project, I have learn more of the uses of progressions in ourdaily life. Apart from that, I have also learn how to bake a cake with accurate height
and length. Other than that, this project encourages the students to share
knowledge and work together. It also encourage students to gather information,
improve their knowledge of mathematical communication. Last but not least, this
project help me get interested in additional mathematics
After conducting this project, I have managed to learn the techniques of applying
additional mathematics in the real life situations.
Last but not least, I have learned many valuable moral value that I could not learn
from the text books. I have learned to cooperate well, share knowledge and work
together with my group members. I have also learned to gather information,
improve my knowledge of mathematical communication.
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things around us that are relatedto circles or parts of a circle.
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